reasonably good matches with the observedprofiles of several Algol-type binaries. He also used the more extensive radiative transfer code of Ko & Kallman (1994) which treats several atomic species, but further work must be done to improve the efficiency of the calculations. Terrell & Wilson (1993) computed the disk matter distribution and motion. Figure 3.25 shows their result for SX Cassiopeiae disk images at 0.32, 1.11, 3.18, and 7.96 orbital revolutions. Based on these calculations, their objective is to derive observable quantities such as spectralline profiles or polarization properties of photospheric radiation scattered by thedisk. Stellar Winds

In addition to the radiation pressure effects discussed in Sect., additional complicated physics is required in hot binary systems. If the components of a binary are hot and produce a sufficiently large radiation pressure as in some Wolf-Rayet stars [cf. Pollock (1987) and White & Long (1989)], they establish radiation-driven hypersonic counterflowing stellar winds [see, for instance, Castor et al. (1975) or Hearn (1987)]. An X-radiation emitting sheet (Prilutskii & Usov 1976) between both stars can block out light emitted toward the observer by one of the stars and significantly modify the light curve and other binary observables. Interested readers should consult Siscoe & Heinemann 1974 and Campbell (1997).

Kallrath (1991) and Stevens et al. (1992) computed the hydrodynamical properties of such colliding winds and the properties of the contact discontinuity established in binary systems. Although Neutsch et al. (1981) and Neutsch & Schmidt (1985) use only simple models for the interface in the context of binary star analysis and compute the effect of this boundary layer on line profiles in HD 152270, their approach is nevertheless instructive.

In hot binaries the opacity increases at the wind interface so as to affect the radiative flux between components. The interface might be considered an attenuating region and could be incorporated into the model as described in Sect. Attenuating Clouds

As in Wilson (1998, 1999) let us use the term "cloud" (more precisely defined, below) to refer to a circumstellar light-attenuating gas or dust region. The attenuation might be due to Thomson scattering, to scattering with (arbitrary) power law wavelength dependence (such as Rayleigh scattering), to continuum opacities, and true absorption, e.g., discrete wavelength absorption features as ultraviolet Balmer lines. Circumstellar matter follows dynamical trajectories, so we might expect there to be little effect from attenuating regions that are fixed in a coordinate frame that rotates with the binary, but such is not entirely the case. A few binaries [e.g., RZ Sct, AX Mon (Elias et al. 1997)] have approximately stationary loci of circumstellar gas that extinct light and distort the light curves. A boundary layer produced by colliding stellar winds in hot binary systems is another example. In general, the loci may be stream-stream, stream-disk, or stream-wind interaction regions. Efforts to represent such light curve distortions via bright or dark star spots sometimes rule out a spot explanation and point to essentially fixed attenuation regions (hereafter, "clouds,"for brevity). The Wilson-Devinney model includes nc spherical semi-transparent clouds specified by their locations (x, y, z) (in a rectangular frame that corotates with the stars; coordinate frame C1 defined in Sect. 3.1.1), cloud radius r, density p, electron density ne, and mean molecular weight per free electron pe. The part of the line-of-sight that passes through the various clouds is computed individually for the lines-of-sight to all surface points and individually for all clouds. Regions of variable density can be made by nesting individual clouds. Regions of nonspherical shape can be approximated by overlapping spherical clouds. Each cloud is allowed its own attenuation law, 41 whose general form is dT

ds where t is the optical thickness, cte is the Thomson scattering cross-section per electron, s is the distance along the line-of-sight (in cm), ka is a wavelength-dependent opacity, and Ksb is an additional opacity for a specific passband (k in cm2/g). The Ksb term might represent, for example, opacity due to absorption lines averaged over a particular passband. The term is ka = K0Xa, (3.4.11)

where k0 and a are input quantities. Each cloud has its individual k0, a, and Ksb. However, to make it easy to change the Ksb of all clouds together, the Ksb's are not entered directly as individual members. Instead we enter an overall Ksb and the fractions fc that applies for each cloud. Thus

where all the fc can be unity if Ksbc is to be the same for all clouds c, or non-unity if Ksbc is to differ from cloud to cloud. The model computes absolute lengths from the system geometry, including the orbital semi-major axis. A first application is to AXMonocerotis (Elias et al. 1997).

3.5 Modeling Radial Velocity Curves

Motu proprio (By one's own motion)

Radial velocities usually are extracted from spectra taken at modest spectral dispersions (1-3nm/mm of reciprocal linear dispersion) through the averaged measure

41 At present the clouds only attenuate starlight that pass through them, but they may be made to scatter starlight toward the observer in a future program version.

ments of absorption line shifts on CCD images. Since they first came into use, the physical size of CCD chips has restricted their wavelength registration range, and as a consequence, velocities have been based on fewer lines than was the case with photogravarPhic or even Reticon detectors of the more recent past.

A powerful means of acquiring good spectral resolution over a large spectral range is through Echelle spectra. These spectra have a large number of orders, each of which has a short spectral range. A cross-disperser is needed to separate the orders. Both background subtraction and ghost images require extra attention in processing, but the advantages of this two-dimensional spectroscopic technique outweigh the disadvantages if enough flux is available. A frequently used method is cross-correlation (see Sect. 2.2.1) of the program star spectrum against that of a velocity standard, with due allowance taken for the reduction of velocities to the Sun. The resulting radial velocities indicate the Doppler shifts of one or both components. Whether Doppler shifts of both components are visible depends on the relative brightness of the less luminous component.

Proximity effects in close binary systems distort not only light curves but also radial velocity curves. The curves are affected by a star's nonsphericity, surface brightness distribution, line strength variation over the surface, aspect dependence of spectral line strength, and eclipses. In binaries with strong tidal distortions or reflection effect heating, these effects have to be accounted for when estimating masses or other quantities derived from velocities.

Even in the case of rotating spheres in a binary the eclipse of part of one of the components results in a phase-dependent shift of the estimated velocity and reflects the dominance of one of the eclipsed star's limbs. In particular, the rotation of the partially eclipsed component distorts the velocity curve. Schlesinger (1909, p. 134) gives already a clear explanation of this "rotation effect": "The rotation of the bright star has another consequence in certain parts of the orbit. In general we obtain light from the whole disk and the observed velocity is equal to that of the center of the star. Just before and just after light minimum, however, this is not the case; before the minimum the bright star is moving away from us and part of its disk is hidden by the dark star. The part that remains visible has on the whole an additional motion away from us on account of the rotation; the observed velocity will therefore be greater than the orbital. On the other hand, just after minimum the circumstances are reversed so that the observed velocity is less than the orbital." Now this effect is called the Rossiter effect42 after Rossiter (1924).

Radial velocity measurements differ from those expected for point sources in other ways as well. That is illustrated in Fig. 3.26 which shows the radial velocity

42 Rossiter also uses the term "rotational effect." For ft Lyrae he measured an amplitude of 13 ± 2km/s and in his paper he wrote: "This is called rotational effect This is, I believe, the first time that this rotational effect has been isolated and measured and eliminated from the least-squares adjustment of the elements. Professor Schlesinger has suspected it in & Librae (he referred directly to the page 134 of Schlesinger's paper) and "


Fig. 3.26 Modeling of the Rossiter effect in AB Andromedae. This plot, produced with Binary Maker 2.0 using the parameter file aband in the examples collection, shows the radial velocity curves for a point source (solid line) and a distorted binary (dotted line)


Fig. 3.26 Modeling of the Rossiter effect in AB Andromedae. This plot, produced with Binary Maker 2.0 using the parameter file aband in the examples collection, shows the radial velocity curves for a point source (solid line) and a distorted binary (dotted line)

curves of the W UMa-type43 EB AB Andromedae. The solid line shows the radial velocity curve we would expect if both stars were point sources. The dotted line is the one including proximity effects. Note the great difference near eclipse phases.

Any program capable of computing photometric light curves should be able to compute radial velocity curves, with only slight extra programming. Intuitively, we expect that measured radial velocities do not correspond to those associated with point masses moving on Keplerian ellipses but rather to the "centers-of-light." The effective radial velocities are the average radial velocities weighted with the local intensities over the stellar surface.

In order to derive results which are independent of the period P and semi-major axis a, the radial velocities may be treated as follows. For radial velocity curves produced by a pair of point masses (see Fig. 3.27) it has been common practice to consider

sin i M1 a2 K2

where the undistortedradial velocity amplitudes K1 and K2 directly give the spectroscopic mass ratio qsp. The quantity w (not to be confused with the argument of peri-astron) is the time-averaged angular velocity of the orbital motion. If we consider deformed stellar surfaces and compute the radial velocity contribution from each surface element, it is not appropriate to keep K1 and K2 in the analysis because they are not uniquely defined in the presence of proximity effects. The natural parameters

43 The Rossiter effect is very significant in strongly distorted eclipsing binaries with short periods as is the case in W UMa-type stars.

Fig. 3.27 Radial velocity curves for point masses. This figure (Courtesy J. D. Mukherjee) shows the radial velocities for a binary system with two point masses

to replace K1 and K2 are the directly physical parameters a1 and a2. An arbitrary surface element da at position r = r(r ; A, v) on the surface of the primary component, seen from the direction s = (sx, sy, ), produces the contribution V (da ) to the radial velocity curve

where vc is the (dimensionless) constant radial velocity of the center-of-mass of the primary, and v is the local radial velocity of da in a corotating coordinate system with origin in the center-of-mass of the primary component. Note that velocities are given in units of am. In those units, in a rotating coordinate frame centered at component 1, v is given by [compare Wilson & Sofia (1976, pp. 183-184)]

where r is the distance from the origin, and T, sx, and sy are the direction cosines defined in (3.1.1) and (3.1.15 ). The effective radial velocity is now computed by averaging over the surface, weighted by the intensity IT of da in the direction s to the observer. A finite grid-point value is computed for each phase 0:

f ITvda

/ It da where / ITda is the integrated flux 1(0) at phase 0. This yields the radial velocity curve V(0):

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